My previous desktop configuration was an Sapphire X1950GT connected to my LCD TV with a DVI/VGA cable. With this setup I also had to have an audio cable running in parallel from the I/O audio to the TV to hear sound from the computer.

However, I have just bought a Sapphire HD6850 which also has an HDMI socket and I was wondering if the HDMI cable will also transmit audio and therefore I can do away with the separate audio cable?

I presume it will, since the link cable for my Panasonic camera (HDMI to HDMI Mini) does convey audio to my TV.

You will also need to make certain that the ATI HDMI sound device is selected. If you have a sound card or a mobo with onboard sound enabled, that device will also be present and you'll need to tell the OS which sound device to use. In this case, that would be the ATI HDMI sound device.